Singularities and full convergence of the M\"obius-invariant Willmore flow in the $3$-sphere

TL;DR

This paper studies the behavior and convergence of the M"obius-invariant Willmore flow in the 3-sphere, analyzing limit surfaces, singularities, and conditions for full convergence to the Clifford torus.

Contribution

It provides new insights into the limit behavior of the MIWF, including the construction of divergent flow lines and criteria for convergence to the Clifford torus.

Findings

Limit surfaces can be identified with integral 2-varifolds.

Flow lines with initial energy ≤ 8π tend to converge to the Clifford torus.

Under certain conditions, parametrizations of limit surfaces are diffeomorphisms of class W^{4,2}.

Abstract

Here we continue the investigation of the M\"obius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus $\Sigma$ into $\mathbb{R}^n$ respectively $\mathbb{S}^n$. Here we investigate the behaviour of flow lines $\{F_t\}$ of the MIWF in $\mathbb{S}^3$ starting with relatively low Willmore energy, as the time $t$ approaches the maximal time of existence $T_{max}(F_0)$ of $\{F_t\}$. We succeed to construct divergent flow lines, and we investigate limit surfaces of both divergent and convergent flow lines of the MIWF. At least generically a limit surface of some general flow line $\{F_t\}$ of the MIWF can be identified with the support of an integral $2$-varifold $\mu$ in $\mathbb{R}^4$, which is the weak limit of the sequence of varifolds $\{\mathcal{H}^2\lfloor_{F_{t_{l}}(\Sigma)}\}$, for an appropriately chosen sequence $t_{l} \nearrow T_{max}(F_0)$, and that $spt(\mu)$ is either empty or homeomorphic to some compact, closed manifold of genus either $0$ or $1$. In the particular case in which $spt(\mu)$ is a compact surface of genus $1$ it can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism $f$ of class $W^{2,2}\cap W^{1,\infty}$, and under certain additional conditions on $\{F_{t_{l}}\}$ such a parametrization is a diffeomorphism of class $W^{4,2}$. Finally, if the initial immersion $F_0$ of a flow line $\{F_t\}$ is assumed to parametrize a Hopf-torus in $\mathbb{S}^3$ with Willmore energy not bigger than $8 \pi$, then we obtain more precise statements about the flow line $\{F_t\}$ as $t \nearrow T_{max}(F_0)$. This insight will finally yield a criterion for full convergence of such flow lines of the MIWF to parametrizations of the Clifford torus - up to M\"obius-transformations of $\mathbb{S}^3$ - as $t \nearrow \infty$.